Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#2176 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4);(5,6,8,7)(9,10)(11,19)(12,23)(13,28)(14,16)(15,20)(17,26)(18,25)(21,39)(22,36)(24,34)(27,35)(29,33)(30,38)(31,40)(32,37);(9,11,14,22)(10,15,16,30)(12,24,27,40)(13,29,25,21)(17,33,32,39)(18,34,28,31)(19,35,36,23)(20,37,38,26);(5,7,8,6)(10,16)(15,30)(17,32)(18,28)(19,36)(23,35)(31,34)(33,39)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400430);